Making Real Analysis Relevant to Secondary Teachers: Building Up from and Stepping Down to Practice

Abstract

Future teachers often claim that advanced undergraduate courses, even those that attempt to connect to school mathematics, are not useful for their teaching. This paper proposes a new way of designing advanced undergraduate content courses for secondary teachers. The model involves beginning with an analysis of the curriculum and practices of school mathematics and its teaching, and then using those to build up to the advanced mathematics – in this case, real analysis. After developing definitions, examples, theorems, and proofs, the model then reconnects to practice, asking the teachers to translate ideas from real analysis in ways that are appropriate for teaching high school content to students. To illustrate the model, we provide and discuss two example tasks.

Keywords:

• Real analysis
• secondary teacher preparation
• mathematical knowledge for teaching
• pedagogical content knowledge

FUNDING

This material is based upon work supported by the National Science Foundation under collaborative grants DUE 1524739, DUE 1524681, and DUE 1524619. Any opinions, findings, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Notes

¹ Although we use “PST” throughout this paper to simplify terminology, we also imagine in-service teachers (ISTs) benefitting from such an approach as well.

² We excluded “n ≠ 0” at the end of the rule so as not to immediately lead PSTs away from considering the number sets Z, Q, and R. This constraint will serve as a point of discussion with PSTs in the next part of the task.

Additional information

Funding

This material is based upon work supported by the National Science Foundation under collaborative grants DUE 1524739, DUE 1524681, and DUE 1524619. Any opinions, findings, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Notes on contributors

Nicholas H. Wasserman

Nicholas H. Wasserman is an assistant professor of mathematics and education at Teachers College, Columbia University. His scholarly interests focus on teacher development and teachers’ mathematical content knowledge, in particular how more advanced mathematics informs teaching practice.

Timothy Fukawa-Connelly

Timothy Fukawa-Connelly is an assistant professor in Temple University’s College of Education. He previously spent 5 years teaching mathematics at the University of New Hampshire and 3 years teaching high school mathematics in Cambridge, MA.

Matthew Villanueva

Matthew Villanueva earned his Bachelor of Arts degree in Mathematics from Rutgers University in 2011. He was a part-time lecturer at Rutgers before joining their Ph.D. program in Mathematics Education in 2013.

Juan Pablo Mejia-Ramos

Juan Pablo Mejía-Ramos is an associate professor at Rutgers University, where he teaches in the Department of Mathematics and the Graduate School of Education. His research focuses on the learning and teaching of mathematics at the university level.

Keith Weber

Keith Weber is a professor of mathematics education at Rutgers University. His research and teaching has focused on reasoning and proof in advanced mathematics.